The reader is directed to [1]} where the expressions of \(\alpha _d\)  and \(\alpha _j\)  (\(j=0,\dots ,3\) ) can be found.
Note that \({\rm H}\partial _x = |D|\)  for the nonlocal term in (REF ).
For the purpose of comparing with the full system (REF )–(),
we have also re-expressed Curtis et al.'s model in a fixed reference frame,
hence the additional advection term in (REF ) as compared to Eq. (2.37) in [1]}.
In the following discussion, we will refer to (REF ) as the “classical” Dysthe equation for this problem,
because it is not Hamiltonian and has the same typical form as in the irrotational case.
Furthermore, its derivation is based on a perturbative Stokes-type expansion for the dependent variables
\(\eta \)  and \(\xi \) , which is similar to the classical derivation by the method of multiple scales [3]}.
Indeed, following [1]}, the surface elevation and velocity potential at any instant \(t\) 
can be reconstructed perturbatively from \(\eta _1\)  as
\( \eta (x,t) & = & \gamma d_1 |\eta _1|^2 + 2 \, {\rm Re} \big ( \eta _1 e^{i \theta } + \ell _0 \eta _1^2 e^{2 i \theta } + \dots \big ) \,,\nonumber \\\xi (x,t) & = & {\rm H}\left( \frac{\Omega _0}{k_0} \eta - \frac{\Omega _0}{k_0} \eta {\rm H}\partial _x \eta - \frac{\gamma }{2} \eta ^2 + \frac{\Omega _0}{2 k_0} {\rm H}\partial _x \big ( \eta ^2 \big ) + \dots \right) \,,\) 
